direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5×Dic5, C5⋊2C20, C10.C10, C52⋊5C4, C10.4D5, C2.(C5×D5), (C5×C10).1C2, SmallGroup(100,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5×Dic5 |
Generators and relations for C5×Dic5
G = < a,b,c | a5=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Permutation representations of C5×Dic5
►On 20 points - transitive group 20T25
Generators in S20
(1 5 9 3 7)(2 6 10 4 8)(11 17 13 19 15)(12 18 14 20 16)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 16 6 11)(2 15 7 20)(3 14 8 19)(4 13 9 18)(5 12 10 17)
G:=sub<Sym(20)| (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,16,6,11)(2,15,7,20)(3,14,8,19)(4,13,9,18)(5,12,10,17)>;
G:=Group( (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,16,6,11)(2,15,7,20)(3,14,8,19)(4,13,9,18)(5,12,10,17) );
G=PermutationGroup([[(1,5,9,3,7),(2,6,10,4,8),(11,17,13,19,15),(12,18,14,20,16)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,16,6,11),(2,15,7,20),(3,14,8,19),(4,13,9,18),(5,12,10,17)]])
G:=TransitiveGroup(20,25);
C5×Dic5 is a maximal subgroup of
C52⋊3C8 Dic5⋊2D5 C5⋊D20 C52⋊2Q8 D5×C20 C52⋊2Dic3 He5⋊5C4 C50.C10 He5⋊6C4
C5×Dic5 is a maximal quotient of
He5⋊5C4 C50.C10
40 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 20A | ··· | 20H |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | D5 | Dic5 | C5×D5 | C5×Dic5 |
| kernel | C5×Dic5 | C5×C10 | C52 | Dic5 | C10 | C5 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 2 | 2 | 8 | 8 |
Matrix representation of C5×Dic5 ►in GL2(𝔽11) generated by
| 9 | 0 |
| 0 | 9 |
| 8 | 0 |
| 0 | 7 |
| 0 | 10 |
| 1 | 0 |
G:=sub<GL(2,GF(11))| [9,0,0,9],[8,0,0,7],[0,1,10,0] >;
C5×Dic5 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_5 % in TeX
G:=Group("C5xDic5"); // GroupNames label
G:=SmallGroup(100,6);
// by ID
G=gap.SmallGroup(100,6);
# by ID
G:=PCGroup([4,-2,-5,-2,-5,40,1283]);
// Polycyclic
G:=Group<a,b,c|a^5=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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